Testing for constant nonparametric effects in general semiparametric regression models with interactions

We consider the problem of testing for a constant nonparametric effect in a general semiparametric regression model when there is a potential for interaction between the parametrically and nonparametrically modeled variables. The work was originally motivated by a unique testing problem in genetic epidemiology (Chatterjee et al., 2006) that involved a typical generalized linear model but with an additional term reminiscent of the Tukey 1-degree-of-freedom formulation, and their interest was in testing for main effects of the genetic variables, while gaining statistical power by allowing for a possible interaction between genes and the environment. Later work (Maity et al., 2009) involved the possibility of modeling the environmental variable nonparametrically, but they focused on whether there was a parametric main effect for the genetic variables. In this paper, we consider the complementary problem, where the interest is in testing for the main effect of the nonparametrically modeled environmental variable. We derive a generalized likelihood ratio test for this hypothesis, show how to implement it, and provide evidence that our method can improve statistical power when compared to standard partially linear models with main effects only. We use the method for the primary purpose of analyzing data from a case-control study of colorectal adenoma.